Allocation of Resources in CB Defense: Optimization and Ranking. NMSU: J. Cowie, H. Dang, B. Li Hung T. Nguyen UNM: F. Gilfeather Oct 26, 2005

Transcription

1 Allocation of Resources in CB Defense: Optimization and Raning by NMSU: J. Cowie, H. Dang, B. Li Hung T. Nguyen UNM: F. Gilfeather Oct 26, 2005

2 Outline Problem Formulation Architecture of Decision System Optimization Methods Raning Procedures Conclusions and Future Wor

3 Problem Formulation Given $M, let θ=(θ,,θ ) be the mitigating variable of some asset, find optimal allocations (x,,x ) to (θ,,θ ) to minimize consequences c=(c,,c ) of an CB attac to the asset, and ran these allocations according to various possible preferences of decision-maers. $M x x 2 x θ θ 2 θ i= x i = $ M

4 Problem Formulation Example: Asset: An airbase Money: $M=million Defense Measures: Consequences: θ : chemical agent detector θ 2 : biological agent detector θ 3 : perimeter protection θ 4 : trained onsite personnel θ 5 : chemical prophylaxis θ 6 : biological prophylaxis θ 7 : medical treatment c : number of casualties c 2 :cost of remediation c 3 :number of days of operation disruption c 4 :negative geo-political impacts

5 Problem Formulation Optimal Allocations We need to formulate an objective function. ϕ : Ω Ψ Ω = = { c Ψ, where {( x =, L ( c Ω is the space of allocations, Ψ is the space of consequences., x, L ) :, c m i = )} x i The optimization problem is Min ϕ(x,,x ) subject to = R M m }, which is an optimization problem with multiple objectives. In principle, the problem can be solved by standard techniques using decision-maers preferences and value trade offs. R ( x, L, x ) Ω

6 Problem Formulation How to obtain the objective function ϕ: R R m? X x, L, x ) θ = ( θ, L, θ ) c( θ ) = ( c ( θ ), L, c ( θ )) = ( m The relation between X=(x,,x ) and the consequence ϕ(x) is: So, a) c(θ): the consequence c is a function of θ. (Data/Scenarios from experts). b) θ(x): is a function of X. (Cost model). ϕ( X) = coθ( X)

7 Problem Formulation Example : An Example of Objective Function c i ( X ) p = α { c s= 0 i ( s) L( s) Π[ e( s, θ ( X ))]} j= j Where: α is a normalization constant, *p is the number of scenarios, *c 0 i (s) initial consequence before improvements for the sth scenario, *L(s) is a probability (sums to ), *θ j (X) is the number of detectors of the i th ind, it is a function of X. *e(s, θ j (X) ) is the effectivity of the i th ind of defense measure on the scenario s.

8 Minimize subject to f Problem Formulation 5 6 s x (,,, ) 0 j x x 2 L x6 = [ sin( π )] 5 x 6 i = x i = 00, x i s = j = Using simulated annealing, we get: 0. ( x, x 2, L, x6 ) Minimum ( ) ( ) ( ) ( ) ( ) ( ) ( )

9 Problem Formulation Histogram for optimal allocations (X=$00) $ to unit $ to unit2 $ to unit3 $ to unit4 $ to unit5 $ to unit6 0 alloc alloc2 alloc3 alloc4 alloc5

10 Problem Formulation Component-wise optimization is not a proper solution for our multiple objective problem. Example 2: Minimize ϕ(x,,x ) over a constrained set A with predetermined acceptable consequences levels. A = {( x, L, x ): xi = M and ϕ j( x, L, x j i= ) cˆ }

11 Architecture INPUT - Data from scenario tree: mitigating variables θ's, lielihood, consequences - Total budget $X - Acceptable consequences C ) - Cost model - Utility function C=ϕ(X) - Human interface: questionnaire ANALYTIC TOOLBOX OPTIMIZATION Minimizes utility function C=ϕ(x) subject to: x and X ) C C RANKING Rans allocations using Choquet Integral, AHP, Multi Objective Programming OUTPUT * x * x2 * x 3 M * x * c * c2 M * c m

12 Optimization. Optimization of the problem in Example 2. Minimize total cost M = w i θ i subject to i= for j=,,m, where c j i= e ij θ i cˆ j * is the number of defense measures (mitigating variables) *w i is the cost of improving θ i by one unit * θ i is the improvement in defense measure facilities θ i *c j is the j th consequence component in the initial variant (data) *e ij is the decrease in the result c j of a scenario attac if we increase θ i by one *m is the number of components of a consequence vector after an attac

13 Optimization Let We want to find θ i s such that is minimized. and ˆ = [200,50,29,3], c [3,5,7,27,6], ],..., [ 5 = w w = ) ( ij e = = i i w i M θ = [000,00,60,20], c = θ θ θ θ θ θ

14 Optimization 0 options of ( θ,..., θ 5) with their corresponding amount of money. ( θ,..., θ 5) =(4,,,,), with M=97; ( θ,..., θ 5) =(,2,,,2), with M=99; ( θ θ =(3,2,,,), with M=99; ( θ,..., θ 5) =(9,,,,4), with M=00; ( θ,..., θ 5) =(,,,,3), with M=00; ( θ,..., θ 5) =(3,,,,2), with M=00; ( θ,..., θ 5) =(5,,,,), with M=00; ( θ θ =(0,3,,,2), with M=0; ( θ,..., θ 5) =(2,3,,,), with M=0; θ,..., ) =(6,2,,,5), with M=02; ( θ5

15 Optimization Histogram for the options #of detector #of detector2 #of detector3 #of detector4 #of detector5 0 opt opt2 opt3 opt4 opt5 opt6

16 Optimization 2. Optimization of vector-valued utility function using decision-maers preferences. To optimize ϕ(x)=(c (X),,c m (X)) T where X=(x,,x ), we can use the common method of weighted linear combination of the components c j (X), i.e., optimize j = where w j is the scaling weight for c j. m w j c j ( x)

17 Optimization How to obtain the w j s? Example: Let c (X)=repair cost ($), c 2 (X)=human casualties. The overall utility function is expressed in $. So w =. Decision maers will be ased to express their preferences among consequences leading to the identification of w 2 (Keeney and Raiffa, 993).

18 The weighting method

19 Optimization 3. Some optimization methods a) Genetic algorithms This "evolutionary" type of optimization method is appropriate for non-smooth objective functions. The method is inspired from the reproduction process in biology. This method is designed to optimize an objective function f for which we do not now its analytic expression but, given input θ =(θ, θ 2,..., θ ), the value f(θ) can be found.

20 Optimization b). Stochastic approximation (Robbins and Monro, 95) This method is designed to optimize an unnown function f(θ) when, for specified θ, the value f(θ) can be provided. This can be done by asing experts from MIIS. p Problem: Find a minimum point, θ* R, of a realvalued function f(θ), called the "loss function," that is observed in the presence of noise.

21 Optimization () Finite Difference: The iterative procedure is ˆ + θ = ˆ θ a gˆ ( ˆ θ ) gˆ i ( ˆ θ ) = y( ˆ θ + c a : goes to 0 at a rate neither too fast nor too slow, e i ) 2c y( ˆ θ y(θ): the observation of f(θ), e i : a vector with a in the ith place, and 0 elsewhere, c : goes to 0 at a rate neither too fast nor too slow. Initialize ˆ θ, calculate gˆ ˆ ( θ) and θˆ,, continue this process 2 till θˆ converges to *, which is our optimization solution. θˆ c e i )

22 Optimization = X = C ~ N (0,) ε and Optimize f(θ)=(xθ-c) (Xθ-C)+ε with Finite Difference procedure,we get the result as follows: Example: Let =[ ] = θ θ θ θ θ θ θ θˆ ] [.667 It is close to the result of the case where the observations are not influenced by noise:

23 Optimization.. (2) Simultaneous Perturbation Stochastic Approximation (SPSA) with Injected Noise (Marya and Chin, 200):to obtain global minimum. ˆ = ˆ θ a g ˆ + ˆ ( θ ) ˆ( θ) = (2c ) [ y( θ + c g θ + q ϖ ) y( θ c )] a,c,q :goes to 0 at a rate neither too fast nor too slow; y(.): the observation of f(.); : distributed as Bernoulli (±); w : i.i.d. in N (0,).

24 Optimization of mocup math model using simulated annealing

25 Raning Ran solutions X = (X, X 2 X ) obtained via optimization to find the most "efficient" one (Multicriteria decision maing, with the mitigating variables θ,θ 2...θ as criteria ). Goal: define a total order between alternatives, i.e. define a map ϕ:r R, so that alternative X is preferred to alternative Y if ϕ(y ) ϕ(x). The total order should reflect the degrees of importance of each criterion θ i in its contribution to the "total score" ϕ(x).

26 Raning In our raning problem, the criteria are interactive, e.g. mitigating variables can contribute to damage reduction in combinations. Non-linear aggregation operators are proved more efficient in such situation than linear ones but may be computationally prohibitive. Approximation can be done with a linear aggregation operator based on simple analysis of interaction between criteria.

27 Raning. Raning with AHP Linear aggregation operator based on simple analysis of interaction between criteria, namely, pair-wise comparisons. Uses fuzzy logic to extract degrees of importance between pairs of criteria, and conducts a synthesis of priorities leading to a weighted average operator. Can handle linguistic (qualitative) values of allocations.

28 Raning 2. Raning with Choquet Integral Non-linear aggregation operator (more general and axiomatically justified). Description of Discrete Choquet Integral: Denote T = c, c, L, c ) to be the set of criteria, ( 2 ( x, x2, L, x x = ) to be the evaluations on the subject x. A fuzzy measure on power set 2 T satisfies a. µ ( 0) = 0, µ ( T) = and, b. A B implies µ ( A ) µ ( B) for A, B T. Discrete Choquet integral with respect to the fuzzy measure is given by C µ ( x) = ( x µ i= ( 0) = ( i) x( i ) ) ( A( i) ) with x 0, and A = c, L, c }, where x, L, x ) is raned x, L, x ) in ( i) { ( i) ( ) ( ( ) ( ) ( increasing order, c, L, c } is the subset of criteria corresponding to x, L, x } { ( i) ( ) { ( i) ( )

29 Raning Example: Ran ten 5-component consequence vectors, with the fuzzy measure µ=[ ]: Consequence vectors Choquet integrals Ran Consequenc e vectors Choquet integrals Ran (4,,,,) (3,,,,2) (,2,,,2) (5,,,,) 6.74 (3,2,,,) (0,3,,,2) (9,,,,4) (2,3,,,) (,,,,3) (6,2,,,5)

30 Raning 3. Identification of fuzzy measures Fuzzy measures have to satisfy the monotone constraints. Two methods to identify fuzzy measures µ: i) Supervised learning a) Quadratic programming b) Neural networ ii) Unsupervised learning (Method of Entropy) Viewing criteria as a random vector, and allocations as random sample. Estimating all joint partial density functions. Using entropies of subsets of criteria as fuzzy measure value.

31 Raning a. Identification of fuzzy measures with quadratic programming For the following data x =(.9.7.5) x 2 =( ) x 3 =( ) x 4 =( ) x 5 =( ) x 6 =( ) x 7 =( ) x 8 =(.5.5.) x 9 =( ) Y = (y, y 2,,y 9 ) =( ). By our algorithm, we get µ = [ ] The corresponding quadratic error is

32 Raning b. Identification of fuzzy measures with neural networs Fuzzy measures are set up as weights of a feed forward neural networ, which are found by training the networ with bacpropagation algorithm. (Wang and Wang, 997).

33 Raning Example: identification of fuzzy measures with neural networs Sample Feature Feature 2 Feature 3 Evaluation Our networ produces the following fuzzy measure: µ{}= 0, µ{}= 0.2, µ{2}= 0., µ{,2}=0.3386, µ{3}= 0.399, µ{,3}= , µ{2,3}= µ{,2,3}=, which satisfies the monotone constraints.

34 Conclusions and Future Wor Data structure Experts and decision-maers assistance Optimization methods Raning procedures Some other issues

35 References [] Deb, K. (2002). Multi-Objective Optimization Using Evolutionary Algorithms. J. Wiley. [2] Grabisch, M., Nguyen, H.T. and Waler, E.A. (994). Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference. Kluwer Academic. [3] Keeney, R.L., and Raiffa (993), H. Decisions with Multiple Objectives. Cambridge University Press. [4] Marya, J. L. and Chin, D. C. (200). Global Random Optimization by Simultaneous Perturbation Stochastic Approximation. Proceedings of the American Control Conference [5] Robbins, H. and Monro,S. (95). A stochastic approximation method. Ann. Math. Statist (22), no 3, [6] Saaty Thomas L., (2000), Fundamentals of Decision maing and Priority Theory, RWS publications. [7] Spall, J.C. (992), "Multivariate Stochastic Approximation Using a Simultaneous Perturbation Gradient Approximation," IEEE Trans. Automat. Control, 37, [8] Verias, V., Lipnicas, A. and Malmqvist, K. (2000). Fuzzy measures in neural networ fusion. Proceedings of the 7th International Conference on Neural Information Processing, ICONIP-2000, [9] Yin, G. (999), "Rates of Convergence for a Class of Global Stochastic Optimization algorithms," SIAM J. Optim.,0, [0] Wang, J. and Wang, Z. (997) Using neural networ to determine Sugeno measure by statistics. Neural Networ (0), No,